Theorem dfdm3 4816

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm3	|- dom A = { x | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4638	. 2 |- dom A = { x | E. y x A y }
2		df-br 4006	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1605	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2293	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2198	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   <.cop 3597   class class class wbr 4005   dom cdm 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-br 4006  df-dm 4638

This theorem is referenced by:  csbdmg  4823